Are there several different meanings for the word "constant"?
We have a number like pi, which is the ratio of a circle's circumference to its diameter. Or e, which is the base of natural logarithms. These things can't change (at least not in our geometry).
Then there are constants like Avogadro's Constant, which is defined to be the number of atoms of carbon-12 in 12 grams (1 mole) of carbon-12, and which is determined empirically. How do we know if we've counted atoms correctly? How do we know if we have exactly 12 grams? This seems to be a pretty shaky type of constant.
Then we have numbers like the gravitational constant, or the fine structure constant. Is it correct to call these "constants" in the same sense that pi is a constant? Google "gravitational constant variable" and you will see that at least some physicists take seriously the possibility of G being different in some models of the Universe. I don't know enough about physics to know whether these physicists are legit though...
The fine structure constant is intriguing. It is approximately 1/137.
It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to ? or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
— Richard P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press 1985, p. 129